Rationality problems and conjectures of Milnor and Bloch-Kato

Details Rationality problems and conjectures of Milnor and Bloch-Kato Rationality problems and conjectures of Milnor and Bloch-Kato

2012-03-19

arxiv.org/abs/1203.4022v1

Mathematics

Algebraic Geometry

15 pages; Revised and extended version of http://arxiv.org/abs/1001.4574 v2; Comments welcome!

Scientific paper

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified \'etale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified \'etale cohomology class of lower degree.

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